reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th68:
  E is antisymmetric iff createGraph(V,E) is non-multi
proof
  set G0 = createGraph(V,E);
  hereby
    assume E is antisymmetric;
    then A1: E is_antisymmetric_in field E by RELAT_2:def 12;
    assume G0 is non non-multi;
    then consider e1,e2,v,w being object such that
      A2: e1 Joins v,w,G0 & e2 Joins v,w,G0 & e1 <> e2 by GLIB_000:def 20;
    e1 in the_Edges_of G0 & e2 in the_Edges_of G0 by A2, GLIB_000:def 13;
    then A3: e1 in E & e2 in E;
    per cases by A2, GLIB_000:16;
    suppose e1 DJoins v,w,G0 & e2 DJoins v,w,G0;
      then e1 = [v,w] & e2 = [v,w] by Th64;
      hence contradiction by A2;
    end;
    suppose e1 DJoins v,w,G0 & e2 DJoins w,v,G0;
      then A4: e1 = [v,w] & e2 = [w,v] by Th64;
      then v in field E & w in field E by A3, RELAT_1:15;
      then v = w by A1, A3, A4, RELAT_2:def 4;
      hence contradiction by A2, A4;
    end;
    suppose e1 DJoins w,v,G0 & e2 DJoins v,w,G0;
      then A5: e1 = [w,v] & e2 = [v,w] by Th64;
      then v in field E & w in field E by A3, RELAT_1:15;
      then v = w by A1, A3, A5, RELAT_2:def 4;
      hence contradiction by A2, A5;
    end;
    suppose e1 DJoins w,v,G0 & e2 DJoins w,v,G0;
      then e1 = [w,v] & e2 = [w,v] by Th64;
      hence contradiction by A2;
    end;
  end;
  assume G0 is non-multi;
  then VertexDomRel(G0) is antisymmetric;
  hence thesis;
end;
